General Optics Design Part 2

GENERAL OPTICS DESIGN PART 2

Martin Traub – Fraunhofer ILT [email protected] – Tel: +49 241 8906-342

 Raytracing  Fundamentals  Aberration plots  Merit function  Correction of optical systems  Optical materials  Bulk materials  Coatings  Optimization of a simple optical system, evaluation of the system

Seite 2

• Real paths of rays in optical systems differ more or less to paraxial paths of rays • These differences can be described by Seidel aberrations up to the 3rd order • The calculation of the Seidel aberrations does not provide any information of the higher order aberrations • Solution: numerical tracing of a large number of rays through the optical system • High accuracy for the description of the optical system • Fast calculation of the properties of the optical system • Automatic, numerical optimization of the system

Seite 3

1. The system is described by a number of surfaces arranged between the object and image plane (shape of the surfaces and index of refraction) 2. Definition of the aperture(s) and the ray bundle(s) launched from the object (field of view) 3. Calculation of the paths of all rays through the whole optical system (from object to image plane) 4. Calculation and analysis of diagrams suited to describe the properties of the optical system 5. Optimizing of the optical system and redesign if necessary (optics design is a highly iterative process)

Seite 4

1. In sequential raytracing, the order of surfaces is predefined  very fast and efficient calculations, but mainly limited to imaging optics, usually 100..1000 rays 2. In non-sequential raytracing, each ray may hit (or miss) each surface as often as necessary, and rays can be absorbed, new rays are generated at partially reflecting surfaces…  limited possibilities for optimization, usually > 106 rays Non-sequential: Sequential: Camera lens laser pump cavity

Seite 5

sd1 n2 (n0,1,3=1) sd3 r2

The optical system is defined by a sequence of surfaces (Sf0 to Sfn). Each surface is defined by 1) Curvature 2) Distance to the next surface 3) Semi diameter 4) Index of refraction of the material following the surface

Seite 6

Image Chief ray plane Marginal ray

Object plane • A number of rays is traced through the optical system • Between two surfaces, the rays are straightly propagating • At each interface with a change in index, refraction or reflection occurs • The beam deflection is calculated applying the full law of refraction (the example shows the marginal and the chief ray)

Seite 7

• Image of an 28 object formed 35 by a three-lens system (not 24 well corrected) B( =8)

• Object size: 24 mm

• The ray bundle of each object point is limited by the aperture B

• To describe the system, ray bundles for different object points are calculated (here: +/-12, +/- 6, and 0 mm)

• As the system is symmetrical, rays below the optical axis are usually neglected

Seite 8

  = 0, 6, 12°

 20 B ( = 14)

• If objects located far from the optical systems are imaged, the incoming rays are nearly parallel (e.g. stars) • Therefore, the field of view is defined by the angle formed by the chief rays and the optical axis

Seite 9

• Wave front aberrations (Optical Path Difference: OPD) TRF • Transverse ray aberration (Transverse Ray Fan: TRF)

• Spot diagrams

• Encircled energy plots

• Modulation transfer function (MTF)

• Point Spread Function (PSF)

encircled energy

OPD Spot Seite 10

Raytracing software uses normalized cartesian coordinates rather than polar coordinates (r, ) to define the position of a ray in the entrance pupil. y   sin x Px   RP RP

  cos  y  Py   RP RP R  P x

RP – pupil radius

Raytracing software uses two points to define each ray: one is located in the object plane, and the other is located in the entrance pupil. Entrance Pupil

Tangential Plane Sagittal Plane

Object Plane

entrance pupil: li,j Reference Sphere • Normalized to the

wavelength and Δy composed to a R wavefront field Optical System Image plane W(x,y) (Detector plane)

• Calculation of the spherical reference wave Wsph,Ref • The OPD q(x,y) (wave front aberration) is calculated by

q(x,y) = W(x,y) - Wsph,Ref(x,y) • The transverse ray aberrations can be calculated by: R q R q x   y   n  n  Seite 13 x y

• Uncorrected singled with large spherical aberrations • The complete 2D OPD (wavefront map) as a function

of Px and Py is shown on the right hand side Px Py • Usually only x and y cross sections are given (OPD fan)

Seite 14

• Wave front aberrations (Optical Path Difference: OPD) TRF • Transverse ray aberration (Transverse Ray Fan: TRF)

• Spot diagrams

• Encircled energy plots

• Modulation transfer function (MTF)

• Point Spread Function (PSF)

encircled energy

OPD Spot Seite 17

a b Detector Py ∆y/ 3 plane C040 · y

∆y(b) ∆y(a)

Px,y - Normalized pupil coordinates ∆x, ∆y - Transverse distance from the axis or the chief ray if the ray bundle is tilted to the optical axis The aberration is measured in wavelengths

Seite 18

• Wave front aberrations (Optical Path Difference: OPD) TRF • Transverse ray aberration (Transverse Ray Fan: TRF)

• Spot diagrams

• Encircled energy plots

• Modulation transfer function (MTF)

• Point Spread Function (PSF)

encircled energy

OPD Spot Seite 19

© Fraunhofer ILT Spot diagram Spot diagram Field angle  = 10° • Start: field of rays Image plane in the entrance pupil with a specific pattern • Calculation and plot of all Field angle  = 0° intersection points of rays on the

image plane V:\A991\2004\VORTRAG\VORLESUNG_TOS_I\05_RayTracing\Links\ZemaxBeis piele\SpotDiagram.ZMX • Gives a good overview of the Intersection points aberrations of the of the rays on the  = 0° optical system image plane

Seite 20

• Well suited for 3rd Square distribution order aberrations Square:

• Artifacts Dithered:

• No artifacts Dithered distribution • Difficult to reproduce

Seite 21

• Wave front aberrations (Optical Path Difference: OPD) TRF • Transverse ray aberration (Transverse Ray Fan: TRF)

• Spot diagrams

• Encircled energy plots

• Modulation transfer function (MTF)

• Point Spread Function (PSF)

encircled energy

OPD Spot Seite 22

2R To every ray with index i in the entrance pupil a portion of the overall energy Etot is assigned:

ri RTot

ETot  eri  1 ei = e(ri)  ri 0 2R Ges The energy EEP(R) encircled in the radius R is calculated by radially summarizing: Spot diagram in the

image plane R EEP R   e ri  ri 0

Seite 23

• Focussing of a parallel ray bundle (with aberrations) Image plane V:\A991\2004\VORTRAG\VORLESUNG_TOS_I\05_RayTracing\Links\ • Summation of ZemaxBeispiele\EncircledEnergy.zmx the energy increments in the image plane Geometrical optics Diffraction limit

Ray tracing with aberrations

V:\A991\2004\VORTRAG\VORLESUNG_TOS_I\V01\06_RayTracing\ Animationen\ZemaxBeispiele\EncircledEnergy.zmx Anteil der eingeschlossenen Leistung Anteil der eingeschlossenen

Seite 24

• Wave front aberrations (Optical Path Difference: OPD) TRF • Transverse ray aberration (Transverse Ray Fan: TRF)

• Spot diagrams

• Encircled energy plots

• Modulation transfer function (MTF)

• Point Spread Function (PSF)

encircled energy

OPD Spot Seite 25

Example: Object pattern Image pattern d I  I Mod  Max Min IMax  IMin

I Max IMax I(x) I(x) I IMin Min 0 0 x x Mod  1 Mod  0,7 Mod MTF( )  Bild  0,7 For a spatial frequency ModObj =1/d in „lines/mm“

Seite 26

Partly corrected focusing triplet with two field angles: 0° (blue) and 3°(green)

No aberrations, no diffraction: Only diffraction, MTF1() MTF() = 1 MTF No aberrations, only Aberrations and diffraction diffraction: for two field angles, MTF1,2() MTF1()

Aberrations and Spatial frequency  (Lines/mm) diffraction:

SeiteMTF 27 1(), MTF2()

• Wave front aberrations (Optical Path Difference: OPD) TRF • Transverse ray aberration (Transverse Ray Fan: TRF)

• Spot diagrams

• Encircled energy plots

• Modulation transfer function (MTF)

• Point Spread Function (PSF)

encircled energy

OPD Spot Seite 28

© Fraunhofer ILT Point Spread Function (PSF) PSF (f=100 mm, EDP=10 mm, The PSF calculates the two- =1 μm): dimensional distribution of intensity in the image plane of the optics for a point source (e.g. a star)

Both diffraction and aberrations are regarded Log scale plot:

For a uniformly illuminated m F μ 

pupil and a system without m aberrations, the PSF can be μ = 24.4 = 2.44

calculated by: 160

Airy

2 d Seite 29

On axis:

If aberrations decreases the optical performance of the system, the diameter of the PSF increases and the maximum intensity decreases: SR=1

, .

Off axis: The Strehl ratio is the ratio of the peak irradiance of an aberrated system to the peak irradiance of the aberration free system. SR=0.4

Seite 30

The main benefit of raytracing software is the ability to optimize the image quality of optical systems automatically (i.e. to reduce the aberrations). The following steps are necessary for this:

1. Declaration of variables Vi of the optical system (e.g. curvatures and distances of surfaces)

2. Definition of target values (e.g. focus diameter or max. OPD)

3. Definition of the merit function (Vi). The merit function calculates a scalar based on the weighted deviation of the target values

4. Finding a local or global minimum of the merit function

Seite 31

Potential variables (indicated by „V“): • Curvature of surfaces • Distances of surfaces • Optical materials • Position and diameter of apertures • Aspherical parameters of surfaces • ......

Seite 32

Typical measures for the degree of modulation for the OPD are:

qPV (Peak-Valley) qcent qRMS qPV r qPV = |qi,Max| - | qi,Min|

qRMS (root mean square) N N 1 1 2 qcentr   qi qRMS   (qcentr  qi ) N i 1 N i 1

Seite 33

Centroid calculated from N spots: 2 RRMS 1 N r centr .   r i N i1

Ri A typical radius of the spot distribution is calculated using the root-mean-square rcentr relative to the centroid:

N ri 1   2 RRMS  (ri  rcentr ) N i1

N 1  2   (ri ) for rcentr  0 N i1 RMS – root mean square

Seite 34

• Definition of parameters j and target values j,target, e.g.:

RMS = RRMS RMS,target = 0 RMS radius of the spot diagram

PV = qPV PV,target = 0 Peak-to-valley modulation of OPD

f/# = f/D f/#,target = 5 F number in image space

• The parameters j depend on the variables Vi • Calculation of the differences between parameter and target value:

j = (j-j,target)

2 • Squaring and weighting of the deviation: wjj • The merit function is the square root of the sum of these values, normalized by the square root of the sum of the weights:

M 1 2 (V )  w  (V ) i  j j i w j j 1 Seite 35

• Initial value V0 is given (V0) • Calculation of the derivative /V at the initial value

• Based on the direction and the  (V1) local Minima increment V, the next value is calculated V global Min. • Recalculation of until the minimum V V 0 1 V is reached

• Mathematical method: (Damped) least squares:  grad ((V ))  0 Seite 36

• Given: merit function, depending on two variables E D C • The contour line plot shows 6 minima (A to F)

• Three weak minima (D to F; trivial F solutions)

• Three strong minima (A to C; desired A solutions) B V2 • Depending on the chosen initial position, different solutions are V1 found

• Evaluation of the solution, choosing

Seite 37 a new initial solution if necessary

Define specification

Select Define variables preliminary and boundary layout conditions

Define merit Evaluate Optimize function

Check availability and price of Tolerance analysis optical components

Mechanical Prototyping design

Seite 38

Optical design • The correction of the aberrations is defined by the parameters chosen during the design process • The sensitivity to tolerances is also defined during the design process Components and • Mechanical tolerances of the optical manufacturing components (curvatures, thicknesses, diameters, centering) • Tolerances of the optical materials (index, dispersion, homogeneity) • Mechanical tolerances of the lens mounts Environment • Temperature range • Mechanical stress • Dynamic loads (vibration and shock)

Seite 39

Initial system for optimization (first layout):

• Catalogs of lens manufacturers (basic designs) or • Databases (patents, LensView, Zebase)

• Experience and intuition

Seite 40

• The map shows well-known lens „Double Gauss“ designs for different combinations of field and speed • The performance and complexity increases from bottom to top Triplet (higher FoV) and from left to right (higher speed)

W. J. Smith: Modern Lens Design, McGraw-Hill

Seite 41

Singlet

• Doublet:

Correction of spherical aberrations and other 3rd order aberrations (limited in field size)

• „double gauss“ lens:

Well corrected for large FoV and low F#

Seite 42

Field flattener

Cooke triplet

Telefocus lens

Retrofocus lens

Seite 43

• The retrofocus lens at the top offers the same effective focal NA1 length (EFL) and NA r compared to the F1 singlet at the bottom

• For the retrofocus A1 lens, the back focal length (BFL) is larger than the EFL NA2 rF2

NA1  NA2 A1  A2

• Also, the telephoto lens has the same EFL NA1 than the singlet (bottom)

• The overall length of the telephoto lens is considerably smaller L1 compared to the singlet NA

NA1  NA2 L1  L2 L2

Approx. 1

Encircled Energy SF6, n=1.7738 NA=0.1 OPD

Approx. 0.6

Radii are chosen to yield the same EFL. Seite 46

Approx. 1

Encircled Energy BK7 NA=0.1 OPD

Nearly diffraction limited. Approx. 0.1

Seite 47

Sphere: Sphere High bending of the peripheral rays (spherical aberration)

Asphere: z

Decreased slope at the edge (weaker refraction) leads to a stigmatic on-axis image Asphere

Seite 48

© Fraunhofer ILT Description of the asphere c  r 2 N z   a r i 2 2  i 1 1 1 K c r i 1    Higher order Conic section terms c – Curvature (=1/R) K – conic constant Surface of K Conic section revolution K 0 Circle Sphere K < -1 Hyperbola Hyperboloid r K = -1 Parabola Paraboloid -1 < K < 0 Ellipse Ellipsoid „Oblate K > 0 z ellipsoid“

Seite 49

Encircled Energy Spherical surface (BK7, NA=0.1): OPD

ca.1

Aspherical surface (BK7, NA=0.1): OPD Encircled Energy K=-0.58  elliptical surface <0,001

diffraction limited

r = 

Seite 50

1. Tilt of lens 2. Wedge error caused by cementing 3. Variation of index 4. Decentered doublet 5. Surface irragulations 6. Striae, Bubbles 7. Lens mount tolerances (tilt) 8. Center thickness deviations 9. Decentered lens 10. Air space tolerances

Seite 51 11. …

• Example: optical system for imaging an optical fiber exit on a workpiece (demagnified, M1:2) • Boundary condition: Working distance > 25 mm, only stock lenses (lead time, costs) • Result: 5-lens-system, retrofocus • Interface Raytracing/CAD: IGES

Seite 52

• All lenses are mounted in a metal cylinder • The distances are defined by spacers • The lenses are centered by the cylinder O-ring O-ring O-ring • Tilt errors are cumulated • To compensate CTE mismatch, elastic rings are foreseen

Seite 53

• Aluminum housing • Three subassemblies Fiber • Flange for mounting the optics to the robot

Flange 50 mm

Seite 54

1 Objective thread 2 Stop face of the objective 3 Spring system for protection mechanism 4..7 Lens groups 8 Correction collar for adapting to cover glass thickness 9 Front lens system 10 Front lens holder

Source: Carl Zeiss AG Seite 55

A) The lens is decentered to the outer cylinder of lens mount B) The mount is tilted and shifted Source: Trioptics GmbH (C1 and C2 coincide with the axis of revolution of the lathe) C) The outer cylinder and the plane surface of the mount are machined

Source: Fraunhofer IPT Seite 56

• The index of refraction of transparent n() materials depends on the wavelength (dispersion, n=n()) • Therefore the axial focal point depends on Thin lens: the wavelength 1  1 1  D   n()  1    • For normal dispersion f  r1 r2  glasses, dn/d<0  n()  1 K • To correct CAs, positive and negative lenses of different index and dispersion are combined (”Achromat“)

The index n is depending on the wavelength n is usually given for specific spectral lines, and a corresponding

suffix is added to the index (e.g. nd)

Symbol Wavelength (nm) Element Spectral region i 365.0 Hg Ultraviolet g 435.83 Hg Blue

F 486.1 H2 Blue e 546.07 Hg Green d 587.56 He Green-Yellow

C 656.3 H2 Red t 1013.9 Hg Infrared

• Mean index of refraction nd (@ d = 587.56 nm)

• Abbe number (@ d line)

nd 1 (for normal dispersion) d   0 nF  nC Large Abbe number: low dispersion

Low Abbe number: large dispersion • Additional data: -Price - Hardness, Machinability - Resistance (Acids, base, water) - Absorption, homogeneity - Thermal expansion and dn/dT

• Optical glass is a inorganic melting product Solidifies as amorphous material Highly transparent, homogeneous and isotropic

• Glass consists of network formers (e.g. SiO2), modifiers (e.g. Ca, Pb) and intermediates • The modifiers change the physical properties of the glass

• Crystals are grown from the liquid phase (very slow, energy-consuming and thus expensive process) • Crystals offer unique features (laser active media, nonlinear properties, anisotropic properties, high thermal conductivity, hardness) • Metals are non-transparent, but offer a high reflectivity and high damage threshold

crown flint glasses glasses

Seite 62

© Fraunhofer ILT Transmission  T  a e n = 1 Transmission: e n e   ' e,R T  a a i Internal Transmission: e ' a,R c0 n    n 12 c  c 2 c a,R  e,R  r 2  R   0 2 2  a e n  1 c0  c  e Reflectivity R: typically <10%, e' e  e,R e' weakly depending on  0' a  a,R (due to dispersion)

 '   'exp   x a' a e

a Coefficient of absorption  x (1/cm) strongly depending on 

 Destructive interference of partial waves which are reflected at the  0 coating-air-interface (1) and at the coating-substrate-interface (2)

Simple case: single layer, 0° A.o.I.

Phase condition:

0 0 n 2n  ds   d  2 2 S 4ns 2 Equal amplitude condition: 1

0 ! ns  n1 n2  ns r01  r12   ns  n1n2 ns  n1 n2  ns

• The beam deviation is P caused by a radial index abs f profile • First example: Thermal L lens of a high power laser L window GRIN lens • Second example: Gradient r index materials

Seite 65

· Folding mirrors • Very high thermal conductivity (kcu 300 kSilica), efficient cooling

• High reflectivity in the NIR spectral range

• Well machinable (UP turning and milling yields to optical surfaces)

• High damage threshold mirrors for high power Off-axis CO2 lasers paraboloid Mirrors of a coaxial ring resonator

Seite 66

• Copper, silver and gold reaches a reflectivity >99% in the FIR region (e.g. 10.6μm) Cu • Copper is commonly used

•Typical absorption: 0.5-2%

Seite 67